Abstract
A topological insulator is a state of quantum matter that, while being an insulator in the bulk, hosts topologically protected electronic states at the surface. These states open the opportunity to realize a number of new applications in spintronics and quantum computing. To take advantage of their peculiar properties, topological insulators should be tuned in such a way that ideal and isolated Dirac cones are located within the topological transport regime without any scattering channels. Here we report abinitio calculations, spinresolved photoemission and scanning tunnelling microscopy experiments that demonstrate that the conducting states can effectively tuned within the concept of a homologous series that is formed by the binary chalcogenides (Bi_{2}Te_{3}, Bi_{2}Se_{3} and Sb_{2}Te_{3}), with the addition of a third element of the group IV.
Introduction
Among the currently known topological insulators (TIs), the layered binary compounds Bi_{2}Te_{3}, Bi_{2}Se_{3} and Sb_{2}Te_{3} have been heavily investigated both experimentally^{1,2,3,4} and theoretically^{5,6,7,8,9}. The realization of many of the predicted topological phenomena requires tuning of the conduction properties of topological insulators, when they are in contact with magnetic semiconductors or superconductors^{10,11,12,13,14,15,16,17,18,19}. This tuning or modification of the topological state is usually achieved by doping or intercalation. A promising way to designing new TI materials with tunable surface electronic properties relies on the fact that pseudobinary systems A^{IV} B^{VI} − m A_{2}^{V} B_{3}^{VI} (A^{IV}=Ge,Sn,Pb; A^{V}=Bi,Sb; B^{VI}=Te,Se) contain homologous series of ternary compounds. These include a wide variety of mixedlayer materials that have a more complex crystal structure than their parent Bi_{2}Te_{3}type compounds. All these systems have tetradymitelike layered structures but with distinct many layered slabs stacked along the c axis of the hexagonal unit cell. The bonding within the slabs is ioniccovalent, whereas the slabs are linked by weak van der Waals forces. The crystal structures of the parent compound and Pbbased systems are shown in Figure 1. In contrast to Bi_{2}Te_{3} (Fig. 1a), whose structure is formed by quintuple layers (5Ls), the structure of PbBi_{2}Te_{4} contains septuple layer (7L) slabs (Fig. 1b), whereas the rest of compounds in this series are formed by combining 5L and 7L building blocks (Fig. 1c,d).
In the present paper, we show that most of the A^{IV} B^{VI} − m A_{2}^{V} B_{3}^{VI} (A^{IV}=Ge,Sn,Pb; A^{V}=Bi,Sb; B^{VI}=Te,Se) compounds are threedimensional (3D) topological insulators. Peculiarities of bulk spectrum of these materials give rise to more complex surface band structure that depends on surface termination, which can be quintuple or septuplelayer terminated. We predict the existence of exotic buried topological surface states (SSs) strongly protected against surface perturbations. A layer resolved spin mapping analysis of the topological states indicates layer variations in the spinhelicity and an oscillatory z spin component behaviour in the Dirac cone. We show that the spin localization varies with energy, allowing for control of the spin properties by varying the chemical potential.
Results
Parent topological insulators
The binary chalcogenides Bi_{2}Te_{3}, Bi_{2}Se_{3}, and Sb_{2}Te_{3} are 3D topological insulators^{1,2,3,4,5,6,7,8,9} with the conducting Dirac SS mostly localized in the outermost 5L block^{6,9}. The surface electronic structure and spatial localization of the Dirac state are shown in Figure 2a for Bi_{2}Te_{3} as an example. The shapes of the abinitio isoenergy contours agree very well with those predicted from a model calculation by Fu^{20} and are thus expected to show an outofplane spin component S_{z}, which increases with increasing warping (Fig. 2b). The model predicts a clockwise rotating inplane spin projection. Our calculations agree with this finding for the subsurface atomic layers of the topmost 5L. However, the rotation direction is reversed at the topmost Te atom Te_{1} (Fig. 2c). This is similar to the oscillatory layer resolved spin–orbit contributions to the total spin splitting in Pb quantumwell states^{21}.
Band structure of the series
The Dirac cone in the binary systems originates from the inversion of the bands forming the gap edges: the conventional sequence of the energy levels is inverted by strong spin–orbit coupling (SOC)^{22,23}. In binary compounds, the bottom of the conduction band in the vicinity of the Γ point is predominantly derived from the states localized at the outermost 5L chalcogen (Te, Se) atoms whereas the top of the valence band is formed by the pnictogen (Bi, Sb) atoms. A similar band inversion is found in PbBi_{2}Te_{4}, with the only difference that the top of the valence band is occupied by both Bi and Pb (Fig. 3a). A more complicated SOCinduced band structure is found in compounds with alternating 5L and 7L slabs, for example PbBi_{4}Te_{7} (Fig. 3b) and PbBi_{6}Te_{10} (Fig. 3c). Here the gap edges can be determined by the states localized on atoms of certain blocks whereas states formed by other blocks lie below or above the gap edges. As will be shown below, this leads to some peculiarities in the formation of the topological state in these compounds. To check the topological character of these materials, we calculated the topological number ν_{0} (refs 23, 24, 25). As all the considered systems have inversion symmetry, we used the method proposed in ref. 23. Our analysis reveals that most of the considered materials can unambiguously be identified as TIs (Table 1).
Surface electronic structure
Surface band structures of PbBiTe and compounds with varying first, second or third elements are given in Figure 4 and Supplementary Figure S1, respectively. Because the crystal structure of PbBi_{4}Te_{7} is constituted by alternating 5L and 7L building blocks (Fig. 1c), the surfaces formed after cleavage of this crystal have two possible terminations. For the 7Lterminated surface (Fig. 4b), the dispersion of the SS is similar to the 5L termination except that the cone is more warped near the conduction band minimum. The spatial distribution of the Dirac state is almost the same as on the PbBi_{2}Te_{4} surface (Fig. 4a); it is localized in the outermost 7L slab. For the 5Lterminated surface (Fig. 4c), the situation is more intriguing; the topological SS is also localized in the topmost 7L block, which is now a subsurface block. Thus, in this system, the Dirac state at the point is buried about 10–25 Å under the surface, which makes this state more protected against possible surface modifications. This localization of the Dirac state in the 7L block follows from the aforementioned peculiarity of the bulk band structure of PbBi_{4}Te_{7}; as one can see in Figure 3b, the gap edges are formed by spin–orbit inverted states, localized at atoms of the 7L block, which form the Dirac cone on breaking translational symmetry at the surface. Furthermore, at the 5Lterminated surface, the surface states inherent to Bi_{2}Te_{3} survive, that is, the unoccupied Rashbatype spinsplit state and the occupied state at −0.2 eV, which becomes the Dirac state in the parent compound. Accordingly, the dispersion and charge distribution (blue lines in Fig. 4c) of this state are identical to the Dirac state in Bi_{2}Te_{3}.
This dependence of the electron distribution depth on the surface termination is a general property of such homologous series. For example, the 7L (Fig. 4d) and 5Lterminated (Fig. 4e) surfaces of PbBi_{6}Te_{10} are similar to the corresponding terminations of PbBi_{4}Te_{7}. Further, the localization of the Dirac state for the two 5L block terminations (5–5L; Fig. 4f) is similar to that in Bi_{2}Te_{3}. However, in contrast to the parent system, the SS on this termination forms an exotic cone with a Dirac point in the local gap of the valence bulk band. This unusual behaviour can be understood from the fact that the occupied states localized at the Bi atoms of the 5L block (Fig. 3c) are not at the gap edge but below the Bi and Pb states of the 7L block, which for this termination lies deep in the bulk and forms the top of the bulk valence band.
ARPES and STM measurements
To study the depth dependence of the SS, we performed highresolution angleresolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM) experiments on PbBi_{4}Te_{7} single crystals, as shown in Figure 5. Compared with the abinitio results from Figure 4b,c, we observe a rigid band shift of about 500 meV, presumably because of bulk and surface defects. Between a binding energy of ~0.3 eV and 0.5 eV, the spectra (Fig. 5a,b) and the constant energy contours (Fig. 5c) only show the topological surface state. At lower binding energies, one observes both the topological state and the bulk states, which can also be identified by their threefold intensity symmetry. Along the direction, the bulk bands cross the topological state, whereas along the direction, the topological surface state remains clearly separated from the bulk bands. At binding energies higher than the Dirac point, the topological state and the bulk bands overlap for a small energy range and, for even higher binding energies (from 0.6 eV), one observes only the bulk bands. A detailed analysis of the band dispersion of the Dirac cone, as exemplified by the constant energy contours in Figure 5d,e and the slightly convex energy dispersion, indicates that the photoemission signal primarily originates from a 5Lterminated surface. On the other hand, the STM data in Figure 5f clearly indicate that both surface terminations are present. Images show an alternating step height of 10 Å and 14 Å that corresponds to the predicted height of the 5L and 7L layers of the PbBi_{4}Te_{7} crystal, respectively. An analysis of surface termination including 18 images with a total area of 212.7 μm^{2} reveals a coverage of about 53% for 7L layers and 47% for 5L layers. Atomically resolved images exhibit a clear hexagon lattice. There is no discernible difference between the atomic resolution images of either the 5L or 7L indicating that, in each case, the STM is imaging the tellurium surface sublattice. This apparent contradiction between the ARPES and STM results might be resolved by considering the defect density at the surface and the fact that the topological state lies beneath the surface for energies well below the conduction band (Fig. 4c) while on approaching the conduction band it shifts to the outermost 5L block (Fig. 5g). The latter fact leads to further increase of the intensity and the linewidth of the Dirac cone at energies at (and above) the conduction band minimum (Fig. 5a) with respect to the lower energies due to defect scattering. In the 5Lterminated areas of the surface, the topological state is located for small k_{} in the 7L subsurface block, protecting this state against scattering at the surface. On the other hand, at the 7Lterminated surface areas defects might push the topological state down to the next subsurface 7L block, as sketched in Figure 5h, where it is no longer accessible by ARPES. This interpretation is corroborated by the absence of the occupied surface state (blue line in Fig. 4c) in the measurements; the topmost layers can not accommodate a surface state; however, because of its protected nature, the topological state shifts down further into the bulk. This confirms the possibility to grow topological insulators with deeplying, selfprotecting and, thus, technologically relevant surface states.
Spinresolved surface band structure
After having verified the electronic nature of the topological state, we address its spin structure, which is directly linked to the topological properties. In Figure 6a, we visualize the calculated layerresolved spin structure of the Dirac state at 50 and 100 meV above the Dirac point in the two topmost blocks. As one can see, the magnitude of the local spin momentum follows the charge density localization of the state shown in Figure 5g. This behaviour may allow for the possibility to tune the spatial localization of the spin transport channel by shifting the Fermi level. The spin structure of the Dirac state at the 5Lterminated surface of PbBi_{4}Te_{7}, as well as at the 7Lterminated surface of this compound and at the surfaces of the other PbBiTe compounds, agrees in all features with that of its parent binary compound. In particular, we find counterclockwise spin rotation in the topmost Te atom regardless of the surface termination, whereas for all other atomic layers it is clockwise.
To determine the net helicity for the complete wave function of the Dirac state, we performed spinresolved ARPES (SARPES) measurements^{26}. Figure 6b,c shows a spinresolved momentum cut along the direction at a binding energy of 10 meV, cutting both through the topological state and the bulk states. By using a vectorial analysis routine^{27}, we can assign a threedimensional spinpolarization vector to each band throughout the constant energy surface, as summarized in Fig. 6d. Even though the exciting photon does not directly couple to the electron spin, the measured polarization does not always directly represent the spin expectation value of the initial state under study due to spindependent matrixelement effects. To confirm that the measured spin polarization of the topological surface state is an intrinsic property of the initial state, whereas the polarization of the enclosed bulk bands is induced by the photoemission process, the photon energy dependence of the polarization was studied. Supplementary Figure S3 shows a pair of spinresolved momentum distribution curve along at the Fermi level measured with 20 eV and 24 eV photon energy. The measured polarization amplitude is strikingly reduced at 24 eV photon energy owing to the larger background as is apparent in the spinintegrated total intensity. Our analysis routine takes the background into account and confirms that neither the direction nor the length of the spin polarization vector of the topological surface state changes significantly on changing the photon energy. On the other hand, the polarization vector of the bulk states is strongly changed as best observed from peaks 2 and 5. The sense of rotation of the topological state is clearly clockwise and the bands are fully polarized, indicating that the spineigenstate of PbBi_{4}Te_{7} corresponds to a positive helicity.
The outofplane spin polarization shows an even stronger difference in the contributions from each atomic layer, both in direction and magnitude. Unfortunately, SARPES is not suited to link the absolute crystal orientation to the calculations, and it is thus not possible to discriminate which atomic contribution prevails. On the other hand, in the SARPES measurements in Figure 6e–h, the threefold symmetry of the outofplane polarization is reproduced including the smooth changes between . It should be noted that, although the outofplane polarization, P_{z}, is related to the warping^{28} and disappears when the warping factor goes to zero (Fig. 2b), the atomic layer contributions show that warping does not directly determine the direction and magnitude of P_{z}.
Discussion
We have shown that, in the homologous series of ternary compounds based on Bi_{2}Te_{3}, Bi_{2}Se_{3} and Sb_{2}Te_{3}, most of the compounds A^{IV} B^{VI} − m A_{2}^{V} B_{3}^{VI} (A^{IV}=Ge,Sn,Pb; A^{V}=Bi,Sb; B^{VI}=Te,Se; m=1—3) are 3D topological insulators. Part of these systems (m=2,3) represent naturally grown superlattices composed of 5L and 7L blocks, which demonstrate much richer physics than the parent TIs. More complex surface electronic and spin structures originate from peculiarities of the bulk spectrum of these materials.
Our findings provide a promising pathway to tune both the electronic and spin properties of a topological insulator, by using the ordered ternary compounds on the basis of wellknown binary TIs. These compounds exhibit a more complicated surface electronic structure and spin behaviour than their binary parents, which provides an efficient way to manipulate both the spin structure and the spatial localization of the conducting state. This subsequently may allow for a variation of the depth of the spin transport channel below the surface. Our spinresolved measurements and atomic layer resolved theory provide a new pathway to the aimed design of spin structure properties in topological insulators.
Methods
Densityfunctional theory
For a complete and reliable description of the electronic structures, we follow a multicode approach, utilizing three different computer codes that are based on densityfunctional theory. The fundamental quantities are consistently reproduced by all codes. The main part of the electronic structure calculations was performed within the density functional formalism implemented in VASP^{29,30}. In this planewave approach, the interaction between the ion cores and valence electrons was described by the projector augmentedwave method^{31,32}. To describe the exchange correlation energy, we used the generalized gradient approximation^{33}. The Hamiltonian contained scalarrelativistic corrections and the spin–orbit coupling was taken into account by the second variation method^{34}. Complementary calculations were performed using the fullpotential linearized augmented planewave method in thinfilm geometry^{35,36} as implemented in the Fleurcode (URL: http://www.flapw.de). The atomresolved spin structures of the topological states were obtained by both VASP and fully relativistic multiplescattering calculations (layer Korringa–Kohn–Rostoker method)^{37}. In the latter case, the Dirac equation is solved from first principles. Experimental lattice parameters were used for calculation of all considered compounds.
Crystal growth
A single crystalline ingot of PbBi_{4}Te_{7} was grown from melt by the vertical Bridgman–Stockbarger method. At first, the polycrystalline PbBi_{4}Te_{7} was synthesized from highpurity (5N) elements in evacuated quartz ampoule at 1,073 K for 8 h mixing incessantly, followed by air cooling. Afterwards, the polycrystalline sample was placed in a conicalbottom quartz ampoule, which was sealed under a vacuum better than 10^{−5} Pa. At the beginning of the growing process, the ampoule was held in the hot zone (~955 K) of a specially designed threezone tube furnace for 24 h for a complete melting of the composition. The charged ampoule moves from the hot zone to the narrow middle zone with the required rate 1.5 mm per hr. The temperature of the middle zone was 855 K corresponding to the crystallization temperature of PbBi_{4}Te_{7}. Finally, the ampoule entered the third zone of the furnace, where the temperature was 773 K. In this way, layered single crystal with average dimensions of 2 cm in length and 0.7 cm in diameter was obtained. The grown crystal was characterized by Xray diffraction using a Bruker D8 ADVANCE diffractometer with Cu–K_{α} radiation. The obtained diffractogram confirmed the formation of singlephase PbBi_{4}Te_{7} crystal that has a hexagonal lattice of the space group P3m1 (Supplementary Fig. S2). By using EVA and TOPAS V3.0 software the unit cell parameters are refined: a=4.418 Å and c=23.948 Å. These parameters are in good agreement with those found in ref. 38.
Scanning tunnelling microscopy
Scanning tunnelling microscopy (STM) experiments were performed in an Omicron multichamber UHVsystem with a base pressure below 1×10^{−10} mbar using a homebuilt variable temperature scanning tunnelling microscope (VTSTM). Both tungstentip and sample were cooled by a Cryovac continuous flow He cryostat to T=30 K. STM topography images were taken in constant current mode at a tunnelling current I_{t} with the bias voltage V_{bias} applied to the sample.
Photoemission experiments
All photoemission experiments were performed at the Surface and Interface Spectroscopy beamline at the Swiss Light Source at the Paul–ScherrerInstitute using a photon energy of 20 eV. The highresolution ARPES data was measured at 10 K with a Scienta R4000 hemispherical analyser with a total energy and angular resolution of 12 meV and 0.1 degrees, respectively. The SARPES data was obtained using the COPHEE spectrometer equipped with two orthogonal Mott detectors operated at 40 kV. The spinresolved data was obtained at 60 K with a total energy and angular resolution of 80 meV and ±0.75 degrees, respectively.
Additional information
How to cite this article: Eremeev, S.V. et al. Atomspecific spin mapping and buried topological states in a homologous series of topological insulators. Nat. Commun. 3:635 doi: 10.1038/ncomms1638 (2012).
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Acknowledgements
We acknowledge partial support by the University of the Basque Country (project GVUPV/EHU, grant IT36607) and Ministerio de Ciencia e Inovación (grant FIS201019609C0201). Financial support from the Hamburgische Stiftung fuer Wissenschaft und Forschung (Cluster of Excellence NANOSPINTRONICS), the Swiss National Foundation, and SPP 1386 “Nanostrukturierte Thermoelektrika” are also gratefully acknowledged. We also thank I. V. Maznichenko for helpful discussions.
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Firstprinciples calculations were performed by S.V.E., T.V.M., Yu.M.K., J.He. and A.Er.. Crystal growth was done by Z.S.A.. M.B.B. and I.R.A. performed Xray diffraction analysis. Photoemission measurements and data analysis were performed by G.L., B.S., L.P., and J.H.D. STM experiments were done by A.Ei., A.A.K., J.Ha., R.W., and O.P. Coordination between photoemssion and STM teams and conception of experiments and research were performed by J.H.D. Coordination between theoretical teams, experimental team from Baku and photoemission group was performed by E.V.C.. Figures were produced by S.V.E., T.V.M., S.S.T., Z.S.A., G.L., and A.Ei. All authors contributed to the discussion. S.V.E., P.M.E., J.H.D., and E.V.C. were involved in project planning. S.V.E., J.H.D., A.A.K., J.He., J.W. and E.V.C. contributed to writing the manuscript.
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Correspondence to Sergey V. Eremeev.
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Eremeev, S., Landolt, G., Menshchikova, T. et al. Atomspecific spin mapping and buried topological states in a homologous series of topological insulators. Nat Commun 3, 635 (2012). https://doi.org/10.1038/ncomms1638
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